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<article class="li"><h3 class="heading">
<span class="type">Item</span><span class="period">.</span>
</h3>
<p><dfn class="terminology">Ratio Test</dfn>   Convergence of a power series can often be determined by the ratio test. If <span class="process-math">\(a_n\neq 0\text{,}\)</span> and if for a fixed value of <span class="process-math">\(x\)</span></p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
\textcolor{black}{\lim_{n\to\infty} \left|\frac{a_{n+1}(x - x_0)^{n+1}}{a_n(x - x_0)^n}\right|}=|x-x_0|\lim_{n\to\infty} \left|\frac{a_{n+1}}{a_n}\right|\textcolor{black}{=L|x-x_0|.}
\end{equation*}
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<p class="continuation">The power series converges absolutely at that value of <span class="process-math">\(x\)</span> if <span class="process-math">\(|x-x_0|&lt;1/L\text{,}\)</span> and diverges if <span class="process-math">\(|x-x_0|&gt;1/L\text{.}\)</span> If <span class="process-math">\(|x-x_0|=1/L\text{,}\)</span> the test is inconclusive.</p></article><span class="incontext"><a href="sec5_1.html#li-6" class="internal">in-context</a></span>
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